Question

求证：1/(1*2*3)+1/(2*3*4)+......+1/(n*(n+1)*(n+2))<1/4

Answer 1

因为1/[n(n+1)(n+2)]
=1/2*{1/[n(n+1)] - 1/[(n+1)(n+2)]}
所以1/(1*2*3)+1/(2*3*4)+......+1/(n*(n+1)*(n+2))
=1/2*{1/(1*2) - 1/(2*3) + 1/(2*3) - 1/(3*4) + …… +1/[n(n+1)] - 1/[(n+1)(n+2)]}
=1/2*{1/2 - 1/[(n+1)(n+2)]}
=1/4 - 1/[2(n+1)(n+2)]显然小于1/4

Answer 2

先变形
1/[n(n+1)(n+2)]=1/2*(n+2-n)/[n(n+1)(n+2)]=1/2*{1/[n(n+1)]-1/[(n+1)(n+2)]}
利用上面的式子就可以对原式进行裂项，抵消大量相同项，
原式=1/2*{1/2-1/[(n+1)(n+2)]}=1/4-1/[2(n+1)(n+2)]
因为n是正的,所以原式自然小于1/4,
命题得证